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1. Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

   https://doi.org/10.4208/cmr.2023-0051  

   Cai, Mingchao; Ju, Guoliang; Li, Jingzhi (May 2024, Communications in Mathematical Research)

   In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions. 

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2. Least squares preconditioning for mixed methods with nonconforming trial spaces

   https://doi.org/10.1080/00036811.2019.1582032  

   Bacuta, Constantin; Jacavage, Jacob (December 2020, Applicable Analysis)

   We consider a preconditioning technique for mixed methods with a conforming test space and a nonconforming trial space. Our method is based on the classical saddle point disccretization theory for mixed methods and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete compatible spaces are provided. For discretization, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We provide approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf− sup and sup− sup constants of the pair of discrete spaces. We focus on applications to elliptic PDEs with discontinuous coefficients. Numerical results for two and three dimensional domains are included to support the proposed method. 

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3. Efficient and effective algebraic splitting‐based solvers for nonlinear saddle point problems

   https://doi.org/10.1002/mma.9665  

   Liu, Jia; Rebholz, Leo G.; Xiao, Mengying (September 2023, Mathematical Methods in the Applied Sciences)

   The incremental Picard Yosida (IPY) method has recently been developed as an iteration for nonlinear saddle point problems that is as effective as Picard but more efficient. By combining ideas from algebraic splitting of linear saddle point solvers with incremental Picard‐type iterations and grad‐div stabilization, IPY improves on the standard Picard method by allowing for easier linear solves at each iteration—but without creating more total nonlinear iterations compared to Picard. This paper extends the IPY methodology by studying it together with Anderson acceleration (AA). We prove that IPY for Navier–Stokes and regularized Bingham fits the recently developed analysis framework for AA, which implies that AA improves the linear convergence rate of IPY by scaling the rate with the gain of the AA optimization problem. Numerical tests illustrate a significant improvement in convergence behavior of IPY methods from AA, for both Navier–Stokes and regularized Bingham. 

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4. Non-stationary Structure-Preserving Preconditioning for Image Restoration

   https://doi.org/10.1007/978-3-030-32882-5_3  

   Dell'Acqua, P.; Donatelli, M.; Reichel, L. (October 2019, Computational Methods for Inverse Problems in Imaging)

   Non-stationary regularizing preconditioners have recently been proposed for the acceleration of classical iterative methods for the solution of linear discrete ill-posed problems. This paper explores how these preconditioners can be combined with the flexible GMRES iterative method. A new structure-respecting strategy to construct a sequence of regularizing preconditioners is proposed. We show that flexible GMRES applied with these preconditioners is able to restore images that have been contaminated by strongly non-symmetric blur, while several other iterative methods fail to do this. 

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5. Inexact rational Krylov subspace method for eigenvalue problems

   https://doi.org/10.1002/nla.2437  

   Xu, Shengjie; Xue, Fei (March 2022, Numerical Linear Algebra with Applications)

   Ye, Qiang (Ed.)

   An inexact rational Krylov subspace method is studied to solve large-scale nonsymmetric eigenvalue problems. Each iteration (outer step) of the rational Krylov subspace method requires solution to a shifted linear system to enlarge the subspace, performed by an iterative linear solver for large-scale problems. Errors are introduced at each outer step if these linear systems are solved approx- imately by iterative methods (inner step), and they accumulate in the rational Krylov subspace. In this article, we derive an upper bound on the errors intro- duced at each outer step to maintain the same convergence as exact rational Krylov subspace method for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the target invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact rational Krylov subspace method is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost. 

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